# External Grants

## Toward Analyses of Mathematics Classroom Discourse at Scale Using Tools from Applied Mathematics. $351,072. Jessica Bishop, Principal Investigator. Mid-Career Advancement Program [MCA], EHR Core Research Building Capacity in STEM Education Research (ECR: BCSER), National Science Foundation, DUE #2218770, 01/2023 – current.

This project uses analytic tools to better understand how classroom discourse affects the teaching and learning of mathematics. The project will focus on two aspects of classroom discourse: how teachers respond to student thinking, and how teachers and students talk about mathematical authority. The project will use quantitative methods to analyze video data from middle-grades mathematics classrooms. Results will advance knowledge and understanding how teachers and students use discourse in mathematics classrooms to create community and build student understanding in mathematics. Additionally, it will investigate how quantitative tools may help researchers systematically analyze classroom discourse with potential to grow to larger-scale research projects.

## Generating a Research-Informed Transition to a Mathematical Proof Curriculum. $600,000. Paul Christian Dawkins, Principal Investigator; Kathleen Melhuish, Kristen Lew, Kyeong Hah Roh, Co-Principle Investigators. Improving Undergraduate STEM Education: Education and Human Resources (IUSE: EHR), National Science Foundation, DUE #2141925, 07/2022 – current.

This project will develop curricular materials for transition to proof courses in partnership with mathematicians. A key focus of the novel curriculum will be learning to read and reading to learn in the context of mathematical proofs. There are three parts to the curriculum. In the first part, students will compare mathematical statements and proofs to learn about mathematical language and abstract the logic of proof. In the second part, students will engage with rich and challenging proofs of non-obvious statements to experience how proof is capable of providing novel and surprising insights. This reading will be scaffolded to support students in learning how to read mathematical proofs for comprehension. In the third part, students will read the mathematical stories of the curriculum authors to humanize mathematics and help broaden their images of what it means to be a mathematician and who gets to be a mathematician.

## Collaborative Research: Middle School Students’ Graphing from the Ground Up (MS-GGU), $293,841. Hwa Young Lee, Principal Investigator; Hamilton Hardison, Co-Principal Investigator. [Collaborative project with Teo Paoletti (PI); total $444,776]. Discovery Research K-12, National Science Foundation, DRL-2200778, 06/2022–current.

The overarching goals of this project are to (a) advance knowledge of middle school students’ developing understandings of graphs “from the ground up” with attention to underlying coordinate systems and frames of reference that comprise the coordinate systems and (b) iteratively design and test tasks that can enhance students’ graph literacy, and thus provide a foundation for their future STEM coursework and careers. MS-GGU uses a design-based methodology in which we will engage in cycles of intervention and revision to develop, test, and refine cognitive models, constructive itineraries, and instructional tasks. Through several iterations of clinical interviews and small-scale teaching experiments in racially, ethnically, and socioeconomically diverse schools, we intend to produce a theory that explains students’ developing graph understandings and accounts for their varied ways of thinking. Through this process, we will iteratively design instructional tasks and task sequences that support students developing understandings for frames of reference, coordinate systems, and graphs in the widely used Desmos platform.

## Comprehending Conditional Claims’ Proofs Organically (Project C3PO). $402,979. Paul Dawkins, Principal Investigator. EHR Core Research, National Science Foundation, DUE-954768, 10/2020 - current

Though engaging in mathematical proof is a central part of an undergraduate mathematics course of study, and the body of research on student learning in this arena is burgeoning, we have no constructivist models of how students understand or come to learn the logic of mathematical proof. As a result, we do not have research-based learning goals for the Transition to Proof courses being offered by many mathematics departments. In this project, we seek to build upon some promising teaching experiments to address this gap. We propose an initial model of students’ Reflection and Abstraction of Proof Structures (RAPS) based on findings from our current studies and propose how we will develop and extend this model through a series of teaching experiments. This work builds upon the success of our earlier work (Dawkins 2017b, 2019; Dawkins & Cook 2016; Dawkins & Roh 2016, 2019; Hub & Dawkins 2018) modeling students’ learning of the logic of mathematical reference, meaning the conditions for when mathematical statements are true and false. At the heart of this approach is helping students apprehend the questions of logic as they abstract their own mathematical activity to address those questions. In this way, we organically develop insights into students’ content-general learning (of logical relations) as it emerges from within their content-specific reasoning (about the specific mathematical relations they reason about).

## Reasoning Language for Teaching Secondary Algebra. $444,474.00. Cody Patterson, Principal Investigator; Emily Bonner, Priya Prasad, Co-Principal Investigators. Discovery Research K-12, National Science Foundation, DRL-1908825, 06/2019 - current.

The Reasoning Language for Teaching Secondary Algebra (ReLaTe-SA) project proposes to study the teaching and learning of algebra in grades 7-9, with a specific focus on the ways in which classroom language explicitly describes properties of and relationships among algebraic objects. The project seeks to investigate the bi-directional relationship between reasoning-rich algebraic discourse and the mathematical meanings students hold for core algebraic concepts such as equations, the equation-solving process, and functions. With a focus on the teacher, ReLaTe-SA will analyze classroom narratives about algebraic concepts and procedures and provide an 80-hour professional development program designed to support teachers in developing stronger explanations of algebraic objects, their properties, and their relationships.

## Using Technology to Capture Classroom Interactions: The Design, Validation, and Dissemination of a Formative Assessment of Instruction Tool for Diverse K-8 Mathematics Classrooms $1,984,657.00. Kathleen Melhuish, Principal Investigator; Ruth Heaton, Eva Thanheiser, Co-Principal Investigators. Discovery Research K-12, National Science Foundation, DRL-1814114, 09/2018 - current.

This project will refine, expand and validate a formative assessment tool called, the Math Habits Tool (MHT) for Kindergarten through 8th grade classrooms. The MHT is intended to capture patterns of in-the-moment teacher-student and student-student classroom interactions in ways that can promote more equitable access to high quality math learning experiences for all students. The tablet or computer-based tool is intended for use with teacher leaders, principals, coaches, and others interested in assessing teacher practice in a formative way. This project will continue the development of the MHT through: (1) the integration of an access component; (2) analysis of videos collected during prior studies covering a diverse set of classrooms across the K-8 spectrum; (2) a validation study using validity-argument approach and (3) the development, piloting, and refinement of professional development modules that will guide math educators, researchers, and practitioners in using the MHT effectively as a formative assessment of instruction.

## CAREER: Scaffolding Strategies for Undergraduate Mathematics Modeling Skills. $954,000. Jennifer Czocher, Principal Investigator. Faculty Early Career Development [CAREER] Program, National Science Foundation, DRL-1750813, 09/2018 - current.

This project studies how undergraduate Science, Technology, Engineering, and Mathematics (STEM) students learn mathematical modeling skills. Mathematical modeling and quantitative reasoning are key problem-solving skills for successful interdisciplinary collaboration. However, typical mathematics classes often focus on solving existing mathematics problems without sufficient attention to important aspects of modeling: formulating mathematical problems to solve and validating the mathematical models that arise. But STEM students do not easily transfer their knowledge of mathematics concepts and procedures to real-world settings. The project will build on the theories and methods developed during her 2018 Australia Endeavour Fellowship at Australian Catholic University to identify scaffolding strategies that best support the growth of STEM students’ modeling skills.

## Orchestrating Discussions Around Proof, $299,574. Kate Melhuish, Principal Investigator; Paul Dawkins, Kristen Lew, Co-Principal Investigators. Improving Undergraduate STEM Education (IUSE), National Science Foundation, DRL-1836559, 11/2018 - current.

The ODAP project consists of a two-phase design-based research study focused on adapting research-based K-12 practices for orchestrating discussion to the new context of an undergraduate proof-based course. The project team will hypothesize, pilot, and refine a model for promoting productive discussion in this context through a series of task-based interviews (Phase 1) and classroom implementations (Phase 2). The analysis will focus on the teaching moves and task components that promote students’ engagement in three key activities connected to proof: comprehending, validating, and constructing. Findings will contribute to the field by testing the transferability of K-12 supports to the undergraduate proof-based setting, thereby expanding knowledge about supporting productive discussions, and contributing refinements and principles specific to this setting. The project will also offer analyses of student-instructor interactions in an authentic proof-based course setting serving to complement the current literature base that primarily leverages clinical settings

## Developing and Validating Proof Comprehension Tests in Real Analysis, $600,000 ($182,414. Sub award to Texas State). Kristen Lew, Kathleen Melhuish, Co-Principal Investigators. Improving Undergraduate STEM Education (IUSE), National Science Foundation, DRL-1821553, 11/2018 - current.

This project will develop and validate eight reliable proof comprehension tests for an undergraduate real analysis course. The generation of proof comprehension tests serves urgent needs for undergraduate mathematics instructors, students in advanced mathematics courses, and mathematics education researchers. For mathematics instructors, the ability to assess students' proof comprehension would improve feedback on the effectiveness of their lectures or reveal aspects of proofs that students find confusing. In this project, the authors will develop and validate short, multiple-choice tests to reliably assess a reader's comprehension of eight proofs in real analysis. For each proof, the authors will first generate open-ended proof comprehension assessment questions using their theoretical model, observe 12 mathematics majors answering these questions individually, and use their responses to create a large repository of multiple-choice assessment items. They will pilot these items with 12 mathematics majors, before having 200 students complete the eight multiple-choice tests. This study of students' proof comprehension (and its relationship to other competencies), as well as the study of the psychometric properties of the developed tests, will make a significant contribution to the study of the construct of proof comprehension in mathematics education.

## CAREER: Characterizing Critical Aspects of Mathematics Classroom Discourse, $672,846.00. Jessica Pierson Bishop, Principal Investigator. Faculty Early Career Development [CAREER] Program, National Science Foundation, DRL-1649979, 05/2012 – 05/2020.

This grant focuses on mathematics classroom discourse in Grades 5–7 classrooms to identify constructs related to productive mathematics discourse (e.g., constructs include responsiveness to students’ mathematical thinking or the connectedness of classroom discourse). The ultimate goal is to help refine current discourse-intense theories of teaching and learning. We are currently working in 20 classrooms to describe whether and how discourse varies across grade level, curricular topic, and teachers, with a goal of developing meaningful metrics that effectively measure specific discursive constructs.

## CAREER: Mathematical Instruction for English Language Learners (MI-ELL), $679,000. Alejandra Sorto, Principal Investigator. Faculty Early Career Development [CAREER] Program, National Science Foundation,DRL-1055067, 07/2011 – 2018.

The main research goal of this project is to empirically estimate whether and which classroom factors contribute to mathematics gains of English Language Learners in Texas schools. The emphasis is on mathematical knowledge for teaching (MKT), knowledge of students as English Language Learners, academic language proficiency in English and Spanish, and the mathematical quality of instruction (MQI) in middle grade classrooms. The main educational goal is to develop professional development and instructional activities for pre-service and in-service teachers of ELL that are research-based and focus on the mathematics needed to teach ELL efficiently. Results of the research study, in particular the classroom practices analyses, will be used to revise and improve the materials to produce a model for professional development and teacher education.

## Enhancing Mathematics Teaching and Learning in Urban Elementary Schools: A Cluster-Randomized Efficacy Trial of a Novel Professional Development Approach, $2,488,354.00. Kate Melhuish, Co-Principal Investigator. Discovery Research K-12 (DRK-12), National Science Foundation DRL-1223074, 2012-2017.

The Enhancing Mathematics Teaching and Learning in Urban Elementary Schools project is working with all teachers in grades three through five mid-sized urban school district in order to test the feasibility and efficacy of the Mathematics Studio Model of professional development. The model requires professional development to occur at the school level involving both teachers and principals. The PD is integrated with instruction. The goal of the project is to improve students' engagement and learning in mathematics by fostering effective instruction. Partners in the project include Teachers Development Group, Portland State University, Texas State University, and Horizon Research. We are using a cluster-randomized research design to examine the efficacy of their model. We are using observational measures to identify successful teaching practices as well as student discourse patterns. We are also studying the fidelity of implementation of the model and are looking for specific variables that may be particularly helpful for students who have not been successful in learning mathematics.

## Dynamic Geometry in Classrooms, $2,090,450. Zhonghong Jiang, Principal Investigator. Discovery Research K-12 (DRK-12), National Science Foundation DRL-0918744, 2009-2016

The project is conducting repeated randomized control trials of an approach to high school geometry that utilizes Dynamic Geometry (DG) software and supporting instructional materials to supplement ordinary instructional practices. It compares effects of that intervention with standard instruction that does not make use of computer drawing tools. The basic hypothesis of the study is that use of DG software to engage students in constructing mathematical ideas through experimentation, observation, data recording, conjecturing, conjecture testing, and proof results in better geometry learning for most students. The study tests that hypothesis by assessing student learning in 76 classrooms randomly assigned to treatment and control groups. Student learning is assessed by a geometry standardized test, a conjecturing-proving test, and a measure of student beliefs about the nature of geometry and mathematics in general. Teachers in both treatment and control groups receive relevant professional development, and they are provided with supplementary resource materials for teaching geometry. Fidelity of implementation for the experimental treatment is monitored carefully. Data for answering the several research questions of the study are analyzed by appropriate HLM methods. Results will provide strong evidence about the effectiveness of DG approach in high school teaching, evidence that can inform school decisions about innovation in that core high school mathematics course.

## Project Z: Mapping Developmental Trajectories of Students' Conceptions of Integers, $1,684,316.00. Jessica Pierson Bishop, Co-Principal Investigator. Discovery Research K-12 (DRK-12), National Science Foundation, DRL-091870, 2009 - 2014.

In this grant, my SDSU colleagues, Lisa Lamb and Randy Philipp, and I studied K–12 students’ conceptions of integers and integer arithmetic. Students’ understanding of integers and integer operations is an area within mathematics education wherein relatively little is known; yet, sound conceptual understanding of integers is a crucial aspect for learning algebra. Based on the analysis of over 250 problem-solving interviews, we developed a framework of problem types and ways of reasoning to describe K–12 students’ conceptions of integers and how those conceptions change over time. By adding to a sparse body of research on an important developmental topic, this project is positioned to make a significant contribution. To date, 14 manuscripts from the larger study have been published or accepted for publication in peer-reviewed journals in mathematics education.